Problem 40 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

PSYCHOLOGICAL TESTING In a psychological experiment, it is found that the proportion of participants who require more than \(t\) minutes to finish a particular task is given by $$ \int_{t}^{+\infty} 0.07 e^{-0.07 u} d u $$ a. Find the proportion of participants who require more than 5 minutes to finish the task. b. What proportion requires between 10 and 15 minutes to finish?

Step-by-Step Solution

Answer

a. About 70.47%. b. About 14.66%.

Step 1 - Understanding the Integral

The integral \(\frac{t}^{+\infty} 0.07 e^{-0.07 u} d u\) represents the proportion of participants who take more than \(t\) minutes to finish.

Step 2 - Solve Part a: Set up the Limits

To find the proportion of participants who take more than 5 minutes, set \(t = 5\). The integral thus becomes: \(\frac{5}^{+\infty} 0.07 e^{-0.07 u} d u\).

Step 3 - Compute the Integral for Part a

The integral of \(0.07 e^{-0.07 u}\) with respect to \(u\) is \(-e^{-0.07 u}\). Evaluate this from 5 to \(+\infty \).

Step 4 - Evaluate the Integral at Infinity for Part a

Evaluate \(-e^{-0.07 u}\) from 5 to \(+\infty\). As \(u \rightarrow\ +\infty \), \(-e^{-0.07 u} \rightarrow\ 0\). Thus, it becomes: \[-e^{-0.07 \cdot 5} - 0\] which simplifies to \[ -e^{-0.35} \].

Step 5 - Compute the Result for Part a

The proportional value is \[-e^{-0.35}\]. Approximating, \(e^{-0.35} \approx 0.7047\). Therefore, the proportion is \( -(-0.7047) = 0.7047 \), or about 70.47%.

Step 6 - Solve Part b: Set up the Limits

For participants taking between 10 and 15 minutes, set the integral's limits from 10 to 15. Compute \( \frac{10}{15} 0.07 e^{-0.07 u} d u \).

Step 7 - Compute the Integral for Part b

Integrate \(0.07 e^{-0.07 u}\), yielding \(-e^{-0.07 u}\). Evaluate from 10 to 15: \(-e^{-0.07 \cdot 15} - (-e^{-0.07 \cdot 10})\).

Step 8 - Evaluate and Simplify the Integral for Part b

Calculate \(-e^{-1.05} + e^{-0.7}\). Approximate the exponentials, \(e^{-1.05} \approx 0.3500 \text{ and } e^{-0.7} \approx 0.4966 \). Thus, \(-0.3500 + 0.4966 = 0.1466 \), or about 14.66%.

Key Concepts

exponential integral

The exponential integral is a mathematical function used in various fields, including psychology, to model situations where events occur continuously and at a rate proportional to their frequency. In the given exercise, we use an exponential integral to find the proportion of participants needing more than a certain amount of time to finish a task.

Here, the integral \(\frac{t}^{+\text{∞}} 0.07 e^{-0.07 u} d u\) measures the proportion of those who take more than time \(t\) to complete it. Evaluating this type of integral often involves exponential decay, where the base of the exponential function is the natural number \(e\).

In the context of the exercise, these integrals assess how likely participants are to require a specific duration to finish the task. Utilizing the exponential integral helps in dealing with distributions over time, especially involving continual processes with decay or growth rates.

The exponential integral appears in psychological testing when predictions or analyses involve time to completion, reaction times, or other time-dependent measurements.

psychological testing

Psychological testing involves measuring various aspects of human behavior and cognition. In this exercise, the focus is on how long participants take to finish a task, represented by the time variable \(t\).

Psychological tests often include tasks designed to measure cognitive abilities, reaction times, and other mental processes. These tests can reveal insights about how individuals think, solve problems, and respond to stimuli. The proportion of participants taking more than a set time to complete a task informs us about their performance and endurance.

By analyzing time data using exponential integrals, psychologists can assess the effectiveness of interventions, compare group performances, and potentially identify factors that contribute to slower or faster completion times. This type of analysis is essential for understanding human behavior in varying conditions and environments.

proportional calculations

Proportional calculations help to determine relationships between quantities where changes in one quantity result in proportional changes in another. In psychology, these calculations are often used to evaluate test results and determine performance metrics.

In the exercise, the proportion of participants needing more than \(t\) minutes is found using integrals that calculate areas under exponential curves. This method allows the expression of complex relationships in a simpler form, explaining how one quantity (time taken) impacts another (proportion of participants).

The steps taken to solve for proportions greater than 5 minutes or between 10 and 15 minutes involve:

• Setting the lower limit of the integral (time threshold)
• Computing the integral using exponential functions
• Evaluating the resulting expressions to find the specific proportions

These proportional calculations are significant in providing insights on the task difficulty and efficiency of participants.

Understanding and applying these concepts help students to accurately interpret psychological testing data, and improve the design and evaluation of assessment tools.